This paper focuses on the continuous assignment problem with the eventual aim to solve scheduling problems with irregular cost functions. It consists of partitioning a region of Rd into subregions of prescribed volumes so that the total cost is minimized. The dual problem of the continuous assignment problem is an unconstrained maximization of a non-smooth concave function. The preemptive variant of the scheduling problem with irregular cost functions corresponds to the one-dimensional continuous assignment problem and a lower bound for the non-preemptive variant can be derived. It is computationally tested in a branch-and-bound algorithm.